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Graphs and Combinatorics

, Volume 35, Issue 2, pp 513–537 | Cite as

Packing Chromatic Number of Subdivisions of Cubic Graphs

  • József Balogh
  • Alexandr Kostochka
  • Xujun LiuEmail author
Original Paper
  • 13 Downloads

Abstract

A packing k-coloring of a graph G is a partition of V(G) into sets \(V_1,\ldots ,V_k\) such that for each \(1\le i\le k\) the distance between any two distinct \(x,y\in V_i\) is at least \(i+1\). The packing chromatic number, \(\chi _p(G)\), of a graph G is the minimum k such that G has a packing k-coloring. For a graph G, let D(G) denote the graph obtained from G by subdividing every edge. The questions on the value of the maximum of \(\chi _p(G)\) and of \(\chi _p(D(G))\) over the class of subcubic graphs G appear in several papers. Gastineau and Togni asked whether \(\chi _p(D(G))\le 5\) for any subcubic G, and later Brešar, Klavžar, Rall and Wash conjectured this, but no upper bound was proved. Recently the authors proved that \(\chi _p(G)\) is not bounded in the class of subcubic graphs G. In contrast, in this paper we show that \(\chi _p(D(G))\) is bounded in this class, and does not exceed 8.

Keywords

Packing coloring Cubic graphs Independent sets 

Mathematics Subject Classification

05C15 05C35 

Notes

Acknowledgements

We thank Sandi Klavžar, Douglas West, and the referees for their helpful comments.

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Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  2. 2.Moscow Institute of Physics and TechnologyMoscow RegionRussian Federation
  3. 3.Sobolev Institute of MathematicsNovosibirskRussia

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